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PREFACE

In 1988 the Design Institute for Physical Property Data of the American Institute ofChemical Engineers established Project 881 to develop a Handbook of Polymer SolutionThermodynamics. In the area of polymer solutions, the stated purposes were: (1) provide anevaluated depository of data, (2) evaluate and extend current models for polymers in bothorganic and aqueous, solvents, (3) develop improved models, and (4) provide a standardsource of these results in a computer data bank and a how-to handbook with accompanyingcomputer software. During the four years of this project most of these objectives have beenmet and the results are presented in this Handbook.There are a number of individuals who deserve special recognition for their contributions to this project. Dave Geveke wrote the liquid-liquid equilibria portions of the text andcreated the LLE data bases. Vipul Parekh wrote the sections on the PVT behavior of purepolymers and developed the pure component polymer data base. Manoj Nagvekar, Vitaly

Brandt, and Dave Geveke developed the computer programs. Gary Barone almost singlehandedly generated the extensive VLE data bases. The help of our undergraduate scholars,John T. Auerbach, Brian Lingafelt, Keith D. Mayer, and Kyle G. Smith, was extremelyvaluable. Technical advice and the basic Chen et al. equation of state program weregenerously provided by Professor Aage Fredenslund of the Technical University of Denmark.Finally, we wish to acknowledge the dedicated service of our secretary, Cheryl L. Sharpe.Throughout the project the Penn State staff was assisted and guided by members ofthe Project Steering Committee. These individuals provided technical advice, critical analysisof the model evaluations and computer programs, additional data references, moral support,and, of course, financial support. Without their generous contributions of time and resources this Handbook would not have become a reality.

D. Procedure: Chen-Fredenslund-Rasmussen Equation of State for Estimating the ActivityCoefficients of Solvents in Polymer Solutions .....................................................................................

E. Procedure: High-Danner Equation of State for Estimating the Activity Coefficient of a Solventin a Polymer Solution ...........................................................................................................................

This page has been reformatted by Knovel to provide easier navigation.

Chapter 1INTRODUCTION

A.

OBJECTIVES OF THE HANDBOOK OF POLYMER SOLUTIONTHERMODYNAMICS

Design and research engineers working with polymers need up-to-date, easy-to-usemethods to obtain specific volumes of pure polymers and phase equilibrium data forpolymer-solvent solutions. Calculations involving phase equilibria behavior are required inthe design and operation of many polymer processes such as polymerization, devolatilization, drying, extrusion, and heat exchange. In addition, there are many product applications requiring this type of information: e.g., miscibility predictions in polymer alloys, solventevaporation from coatings and packaging materials, substrate compatibility with adhesives,and use of polymers in electronics and prostheses. This Handbook of Polymer SolutionThermodynamics contains data bases, prediction methods, and correlation methods to aidthe engineer in accurately describing these processes and applications. This Handbookprovides the necessary background information, the most accurate prediction and correlationtechniques, comprehensive data bases, and a software package for DOS based personalcomputers to implement the recommended models and access the data bases.Generally the preferred data source is experimental measurement. Only in rare casesare prediction methods able to give more accurate estimates than a carefully executedexperiment. Therefore, one of the major objectives of this Handbook is to provide comprehensive data bases for the phase equilibria of polymer-solvent systems and pressure-volumetemperature behavior of pure polymers. Thus, data have been compiled from extensiveliterature searches. These data cover a wide range of polymers, solvents, temperatures, andpressures. The data have been converted into consistent units and tabulated in a commonformat. Methods of evaluating and formatting these data banks have been established bythe DIPPR Steering Committee for Project 881 and the Project Investigators.No matter how broad the scope of the experimental data is, there will always be aneed for data that have not yet been measured or that are too expensive to measure.Another objective of this Handbook is to provide accurate, predictive techniques. Predictivetechniques not only furnish a source of missing experimental data, they also aid in theunderstanding of the physical nature of the systems of interest. The most useful predictivemethods require as input data only the structure of the molecules or other data that areeasily calculated or have been measured. Many of the methods present in this Handbookare based on the concept of group contributions which use as input only the structure of themolecules in terms of their functional groups or which use group contributions and readily

available pure component data. In some cases the users of the Handbook will need tocorrelate existing data with the hope of extending the correlation to conditions not availablein the original existing data. Several correlative methods of this type are included.The current state-of-the-art is such that there are no reliable methods of predictingliquid-liquid equilibria of polymer-solvent systems. Thus, the recommended procedures andcomputer programs included in this Handbook treat only vapor-liquid equilibrium. Adiscussion of the correlation of LLE data is included in Chapter 2.Chapter 2 is an in depth discussion of the various theories important to phaseequilibria in general and polymer thermodynamics specifically. First a review of phaseequilibria is provided followed by more specific discussions of the thermodynamic modelsthat are important to polymer solution thermodynamics. The chapter concludes with ananalysis of the behavior of liquid-liquid systems and how their phase equilibrium can becorrelated.Chapter 3 contains the recommended predictive and correlative procedures for thespecific volume of pure polymer liquids and the calculation of the vapor liquid equilibria ofpolymer solutions. These methods have been tested and evaluated with the data basesincluded in this Handbook.Chapter 4 describes the polymer data bases. This chapter is organized into sectionsdiscussing the experimental methods available for measuring the thermodynamic data ofpolymer solutions with an overview of the advantages and disadvantages of each method.The next section, Data Reduction Methods, describes how the experimental measurementsfrom these experiments can be used to calculate the activity coefficients that are necessaryfor phase equilibria calculations. Finally, a summary of all the systems that are available onthe data diskettes is provided. A user can scan this section or use the computer programPOLYDATA to find if data are available for a particular system.The Computer Programs section, Chapter 5, describes the two primary computerprograms on the diskettes accompanying this Handbook. POLYPROG is a program whichimplements the recommended procedures of Chapter 3. POLYDATA provides an easymethod of accessing the data contained in the many data bases. Chapter 6, contains theAppendices. The sections included are Glossary of Terms, Standard Polymer Abbreviations,Nomenclature, Units and Conversion Factors, and References.

Chapter 2FUNDAMENTALS OF POLYMER SOLUTION THERMODYNAMICS

A. PURE POLYMER PVT BEHAVIOR

Volune

Density (or specific volume) is an essential physical property required either directly inthe design of polymer processing operations or as an input parameter to obtain various otherdesign variables. In injection molding and extrusion processes, the design is based ontheoretical shrinkage calculations. Since these operations are carried out at high pressures,compressibility and thermal expansion coefficients are required over wide regions of pressure,volume, and temperature. The PVT behavior can also be coupled with calorimetric data tocalculate the enthalpy and entropy of the polymers in high pressure operations. Since theseoperations are accompanied by high power requirements, accurate estimates of enthalpies arecritical for an energy-efficient design (lsacescu et al., 1971).Figure 2A-1 shows the dilatometric behavior typically observed inpolymers. The melt region correspondsto temperatures above the melting temperature, Tm/ for a semi-crystalline polymer and to temperatures above the glassequilibriun meltamorphous polymerregiontransition temperature, Tg, for an amorphous polymer. The correlation presented in this Handbook is only for the equilibrium melt region. Correlations of thesemi-crystalline polymerPVT behavior of some polymers in theglassy region are given by Zoller (1989).If one wishes to estimate a specificvolume of a polymer in a solution belowTemperatureTg or Tm, however, it may be better toextrapolate the liquid behavior. Exten- Figure 2A-1. Dilatometric Behavior of Polymers.sive testing of this hypothesis has notbeen done.The experimental technique used to measure the PVT data is based on the Bridgemannbellows method (Bridgemann, 1964). The polymer sample is sealed with a confining liquid,usually mercury, in a cylindrical metal bellows flexible at one end. The volume change of thesample and the confining liquid with changes in the applied pressure and temperature isobtained from the measurement of the change in length of the bellows. The actual volumeof the sample is then calculated using the known PVT properties of the confining liquid. The

accuracy of the apparatus is estimated to be around 0.001 cm3/g which corresponds toapproximately 0.1% for polymer specific volumes. A detailed description of this techniqueis given by Zoller et al. (1976).Often empirical models or correlative equations of state are used to describe the PVTbehavior of polymers (Zoller, 1989). Such correlations are useful in the interpolation andextrapolation of data to the conditions of interest. When an equation of state based onstatistical mechanical theory is used to correlate the data, the resulting equation parameterscan also be used in mixing rules to determine the properties of polymer solutions.A number of models have been developed and applied for the correlation of polymerPVT behavior. One of the first was the purely empirical Tait equation (1888). This equation,originally developed to describe the compressibility of ordinary liquids, has been shown towork well for a wide variety of liquids ranging from water to long-chain hydrocarboncompounds (Nanda and Simha, 1964). This approach has also been successfully applied topolymers (Zoller, 1989). In developing the recommended PVT correlation for this Handbookseveral variations of the Tait correlation, the Flory equation of state (Flory et al., 1964), theSimha-Somcynsky equation of state (Simha and Somcynsky, 1969), and the SanchezLacombe equation of state (Sanchez and Lacombe, 1976) were evaluated. The Tait formgiven in Section 3B yielded errors which were generally an order of magnitude lower than thatfound with the other models. In almost all cases, the average error with the Tait model wasfound to be within the reported experimental error - approximately 0.1% (Zoller et al., 1976).The High-Danner equation of state given in Section 3E can be used to predict thespecific volume of polymers. Parekh (1991) has modified some of the reference volumes inthe model to improve the model's accuracy for pure polymer volumes. The deviations in thesepredictions are generally less than 2%. Additional work needs to be done to establish thereliability and to extend the applicability of the method.

B. PHASE EQUILIBRIA THERMODYNAMICSThe design engineer dealing with polymer solutions must determine if a multicomponent mixture will separate into two or more phases and what the equilibrium compositionsof these phases will be. Prausnitz et al. (1986) provides an excellent introduction to the fieldof phase equilibrium thermodynamics.The primary criterion for equilibrium between two multicomponent phases is that thechemical potential of each component, j j { / must be equal in both phases I and II.P\= //!'

< 2B - 1 >

The phases in the system must be in thermal and mechanical equilibrium as well.T1 I' = T-"'i

< 2B - 2 >

piri

pii= r

(2B-3)

i

In some cases, the chemical potential is not a convenient quantity to calculate forengineering purposes. The fugacity of component i , fj, is defined in terms of the chemicalpotential, JJ1, by the expressionfi = RT In /7j

(2B-4)

Thus, for Equation (2B-1) to be satisfied, the fugacities of component i must be equal in bothphases.f' = f»i'i

(2B-5)

Two methods can be used to calculate the fugacities of a component in equilibrium.The first method requires an equation of state, which can be used with the followingexpression to calculate the fugacity coefficient.

' - * - W T / : [(£],,,„,-^]

dv

-inz

The fugacity is related to the fugacity coefficient byfi = 0,Py,

(2B-7)

Here Q1 is the fugacity coefficient of component i, P is the pressure, and y{ is the mole fractionof component i. Fugacity coefficients are usually used only for the vapor phase, so y{ isusually meant to represent the mole fraction of component i in the vapor phase and X1 isusually reserved to represent the mole fraction in the liquid phase. Equation (2B-6) can beused with any equation of state to calculate the fugacity of the components in the mixturein any phase as long as the equation of state is accurate for the conditions and phases ofinterest. An equation of state that is explicit in pressure is required to use Equation (2B-6).If the equation of state is valid for both phases, then Equation (2B-6) can be appliedto calculate the fugacity in both phases. The isochemical potential expression, Equation(2B-1), reduces to0W = 0J1X111

«B-8)

where xj is the mole fraction of component i in phase I and xj 1 is the mole fraction in phase II.In this terminology X1 represents a mole fraction in any phase which could be liquid or vapor.The major difficulty in using Equation (2B-8) is finding an equation of state that is accuratefor both the liquid phase and the vapor phase.The second approach to phase equilibria is to relate the fugacity of a component in theliquid phase to some standard state fugacity and then calculate the deviation from this

standard state. The fugacity in the liquid phase, f \, is calculated from the activity coefficientof component i, KJ/ and the standard state fugacity, f ° using the expressionf1 JL = yr.ixA.fi ' °|

(2B-9)

The fugacity of component i in the vapor phase is calculated with an equation of state as inthe first case using Equation (2B-7). In this case the isochemical potential expressionbecomes01V = KiXif"

(2B-1O)

Many times the virial equation truncated after the second virial coefficient is used inplace of a more complicated equation of state to calculate the fugacity of the components inthe vapor phase.In the case of liquid-liquid equilibria the activity coefficient expression is usually usedto calculate the fugacity in both of the liquid phases0A1X,' fT,0 1 = iWfKy\ X, Tj "

(2B-11)

If the standard state in both phases is the same, then Equation (2B-11) becomesKW = KW'

<2B-12)

All of the expressions described above are exact and can be applied to small non-polarmolecules, small polar molecules, non-polar polymers, cross-linked polymers, polyelectrolytes,etc. The difficulty is finding correct and accurate equations of state and activity coefficientmodels. Many accurate activity coefficient models have been developed to correlate existingactivity coefficient data of small molecules or to predict activity coefficients given only thestructure of the molecules of interest or other easily accessible data (Danner and Daubert,1989).During the past ten years, the chemical process industry has seen an increase in theaccuracy and range of applicability of equations of state. Equations of state are becoming amore popular choice for computing and predicting phase equilibria. Most of the research onactivity coefficients and equations of state, however, has focused on low molecular weightsystems. Relative to small molecules, polymers and polymer solutions are essentiallyunexplored.

C. MODELING APPROACHES TO POLYMER SOLUTION THERMODYNAMICSAll of the models developed for predicting and correlating the properties of polymersolutions can be classified into two categories: lattice models or van der Waals models. Thesetwo approaches can be used to derive activity coefficient models or equations of state.Activity coefficient models are not functions of volume and therefore are not dependent on

the pressure of the fluid. On the other hand, equations of state are functions of volume, andpressure does influence the results.In both the lattice models and the van der Waals models, the behavior of the moleculesis described as the sum of two contributions. The first contribution assumes that there areno energetic interactions between the molecules; only the size and shape of the moleculesneed to be considered for this part. This is the contribution that would be predominant atvery high temperatures where the kinetic energy of the molecules would be large comparedto any interaction energies between the molecules. This interaction-free contribution isgenerally called the combinatorial or the athermal term. In the case of the van der Waalsmodel, it is frequently referred to as the free volume term.In lattice models each molecule (or segment of a molecule in the case of polymers) isassumed to occupy a cell in the lattice. The arrangement of the molecules or segments isassumed to depend upon only the composition and the size and shape of the molecules. Inthis case, the combinatorial (athermal) contribution is calculated from the number ofarrangements statistically possible in the lattice. This contribution is also referred to as theentropic term.In the van der Waals model the volume in which the molecules can translate isdetermined by the total volume of the system less the volume occupied by the molecules.Thus, the term "free volume." In this part of the treatment of the system intermolecularattractions are not taken into account, so this free volume term is the combinatorial (athermal)contribution.The second contribution in either the lattice or the van der Waals model is thatoriginating from intermolecular attractions. This contribution is commonly referred to as theattractive energy term, the residual term, or the potential energy term. It is also known asthe enthalpic contribution since the differences in interaction energies are directly responsiblefor the heats of mixing. This contribution is calculated by a product of a characteristic energyof interaction per contact and the number of contacts in the system. Van der Waals modelsuse a similar expression for the interaction energy.In some of the more sophisticated models, the concept of local compositions is usedto improve the results. Since the combinatorial contribution is calculated without regard tothe interactions between molecules, it leads to a random arrangement of the molecules. Inreality, the arrangement of molecules in a pure component or a mixture is affected by theinteractions.The concept of local compositions is used to correct the combinatorialcontribution for the nonrandomness that results from these interactions. Local compositionexpressions are a function of the interaction energies between molecules and result in acorrection to the combinatorial called the nonrandom combinatorial. There are several theoriesavailable to calculate the local composition and the nonrandom combinatorial, but the mostwidely used theory is Guggenheim's (1952) quasichemical theory. This terminology is usedbecause of the similarity between the equations in Guggenheim's theory and the relationshipbetween the chemical equilibrium constant and the Gibbs energy in chemical reactionequilibria. The major difficulty with using local compositions in activity coefficient models andequations of state is that the resulting models and calculations are usually quite complex. Theincreased accuracy and more general applicability of the equations, however, is usually worththe increased computational cost.

D. LATTICE MODELS1. Florv-Huqqins ModelFlory (1941) and Muggins (1941, 1942a,b,c) independently developed a theory ofpolymer solutions that has been the foundation of most of the subsequent developments overthe past fifty years. In their work, the polymer-solvent system was modeled as a latticestructure, The combinatorial contributions to the thermodynamic mixing functions werecalculated from the number of ways the polymer and solvent molecules were arranged on thelattice sites. These combinatorial contributions correspond to the entropy of mixing. Implicitin the Flory-Huggins treatment of the combinatorial contributions is the assumption that thevolume of mixing and the enthalpy of mixing are zero. The number of ways these moleculescan be arranged leads to the well-known Flory-Huggins expression for the entropy of mixingin a polymer solution.^§ = -N1 In 0! - N2 In 02K

(2D 1)

'

Here N1 and N2 are the number of solvent and polymer molecules, respectively, and thevolume fractions 01 and 02 are defined by the expressions011

=

NI

(2D-2)

N1-HrN2

rN2022 =^N1-H-N2

(2D-3)

where r is the number of segments in the polymer chain. The activity of the solvent, a-, isgiven byWa 1 ) = ln(1-02) + [l-l|02

(2D 4)

'

Several improvements to Equation (2D-4) have been suggested. Primarily these modificationsinvolve a more exact treatment of the polymer chain in the lattice such as including theprobability of overlapping chains. These improvements are not generally applied in view ofthe approximations inherent in the lattice model of the fluid and the marginal increase inaccuracy resulting from these improvements.Flory (1942) noted that the combinatorial term is not sufficient to describe thethermodynamic properties of polymer-solvent systems. To correct for energetic effects, hesuggested adding a residual term, ares, to account for interactions between lattice sites.In a, = In a™mb + InThe residual term suggested by Flory is

res3l

< 2 °- 5 >

In

res3l

- Xl

(2D 6)

'

where x has become known as the interaction parameter or the Flory-chi parameter. Thecritical value of/ for miscibility of a polymer in a solvent is approximately 0.5. For values of/ greater than 0.5 the polymer will not be soluble in the solvent, and for values of/ less than0.5 the polymer will be soluble in the solvent.The Flory-Huggins combinatorial term with the Flory / residual term has been thecornerstone of polymer solution thermodynamics. It established that the major contributionto the excess Gibbs energy and, hence, the activity in polymer solutions, is entropic unlike theenthalpic effects that dominate small molecular systems. As pointed out by many authors,however, there are deficiencies with the Flory-Huggins model. The most serious of these isthat the lattice model precludes volume changes when the polymer molecules are mixed withthe solvent molecules. Since the total volume that can be occupied in the lattice is a fixedquantity and vacancies are not permitted, volume changes cannot affect the thermodynamicpotential functions such as Gibbs energy, and the model exhibits no pressure dependency.Thus, the model is strictly applicable only to liquids which exhibit no volume change ofmixing. Furthermore, as originally proposed, x was independent of composition andtemperature. In fact, x often shows complex behavior as a function of both of theseindependent variables.2. Solubility Parameters and the Florv-Huqqins ModelIdeal solutions are defined as mixtures that have no volume or enthalpy changes uponmixing, but have an ideal entropy of mixing given byAS m = R J>j In X 1

(2D-7)

stated in another way, in an ideal solution the excess entropy, SE, excess volume, VE, andexcess enthalpy, HE, are all equal to zero.Regular solutions are defined as those solutions that have zero excess volume andexcess entropy changes, but a non-zero excess enthalpy. Polymer solutions are not regularsolutions since mixing a polymer with a solvent leads to a non-zero excess entropy change.Therefore, the excess volume, entropy, and enthalpy are all non-zero for a polymer solution.Nevertheless, the concept of regular solutions and the related solubility parameter have beenused to predict the thermodynamic properties of polymer solutions. The regular solution andsolubility parameter concepts developed by Hildebrand and Scott (1949, 1962) provide ameasure of the interaction energies between molecules. These interaction energies, quantifiedby the solubility parameter, can then be used to estimate the / parameter for a polymersolvent system. The solution properties of the solution are easily calculated once the xparameter is known.The solubility parameter, 6{, is defined as the square root of the cohesive energydensity. The cohesive energy density is the amount of energy per unit volume that keeps thefluid in the liquid state. An excellent approximation for the cohesive energy of a solvent, GJJ,is the heat of vaporization, which is the amount of energy that must be supplied to vaporizethe fluid. The solubility parameter is calculated from

1/2*i = CH=A

AEjVap-^-

(2D-8)

The solubility parameter can be used to estimate the Flory interaction parameter,/, via:X = ^W1 - <52)2nI

(2D-9)

where V1 is the liquid molar volume of the solvent, and (J1 and 62 are the solubility parametersof the solvent and polymer, respectively.As mentioned in the previous section a value of x 'ess than 0.5 indicates that thepolymer will be soluble in the solvent. The smaller the value of / the more soluble thepolymer should be. Thus, from Equation (2D-9) it is clear that the closer in value the twosolubility parameters are, the more compatible the components will be. WhenJ1 = 62

(2D-10)

X is zero and the optimum condition is obtained. Unfortunately, because of the assumptionsin the models, the above criterium should be regarded only as a qualitative measure ofmiscibility.Since the Flory interaction parameter, /, was derived by considering only interactionenergies between the molecules, it should not contain any entropic contributions and Equation(2D-9) should yield the correct value for the Flory-x parameter. Unfortunately, x contains notonly enthalpic contributions from interaction energies, but also entropic contributions. Thesolubility parameter includes only interaction energies and by the definition of regular solutionsdoes not include any excess entropy contributions. Blanks and Prausnitz (1964) point out thatthe Flory x parameter is best calculated fromJT-JT8

+

^W1 -62)2nI

(2D-11)

where the entropic contribution to the / parameter, /s, is given byJf8 = 1

(2D-12)

Here z is the coordination number of the lattice; i.e., the number of sites that are nearestneighbors to a central site in the lattice. Blanks and Prausnitz suggest a value of/ s between0.3 and 0.4 for best results.There are many sources of data for the solubility parameters of solvents and polymers.Daubert and Danner (1990) have compiled accurate solubility parameters for over 1250industrially important low molecular weight compounds. Barton (1983, 1990) has tabulatedsolubility parameters for most of the industrially important polymers.Experimental methods for solubility parameters of polymers commonly involveobserving the swelling of the polymer as solvent is added. After performing this experimentwith a number of solvents with different solubility parameters, the solvent which leads to thegreatest degree of swelling for the polymer is the best solvent for that polymer. Since a xvalue of zero in Equation (2D-9) indicates the degree of solubility of a polymer in a solvent,

the solubility parameter of the polymer is approximately equal to the solubility parameter ofthe best solvent.The problem remains of how to predict the solubility parameter of the polymer givenonly readily available information such as pure component properties or structure. Barton(1983, 1990) and van Krevelen (1990) have proposed group contribution methods that maybe used, but these methods are extremely empirical and give qualitative results at best.One of the major deficiencies with the solubility parameter concept is that onlyinteraction energies arising from dispersive forces are involved in the definition of the cohesiveenergy density. Molecules that are polar or that hydrogen bond cannot be modeled with theHildebrand-Scott solubility parameter. In order to improve the predictive results using thesolubility parameter, Hansen (1969) proposed that the cohesive energy be divided intocontributions due to dispersion forces, permanent dipole-permanent dipole forces, andhydrogen bonding forces. The overall solubility parameter is calculated from the contributionsfrom these three types of interactions.6* = 62d + 62p + 62h

(2D 13)

-

Here 6^1 6p, and £h are the contributions to the solubility parameter from dispersive forces,dipole-dipole forces, and hydrogen bonding forces, respectively. Since the three forces canoccur to varying degrees in different components and can be represented on a threedimensional diagram, this theory is termed the three-dimensional solubility parameter. Barton(1983, 1990) tabulates the contributions to the three dimensional solubility parameter for avariety of solvents and polymers.Regular solution theory, the solubility parameter, and the three-dimensional solubilityparameters are commonly used in the paints and coatings industry to predict the miscibilityof pigments and solvents in polymers. In some applications quantitative predictions have beenobtained. Generally, however, the results are only qualitative since entropic effects are notconsidered, and it is clear that entropic effects are extremely important in polymer solutions.Because of their limited usefulness, a method using solubility parameters is not given in thisHandbook. Nevertheless, this approach is still of some use since solubility parameters arereported for a number of groups that are not treated by the more sophisticated models.3. Modifications of the Florv-Huqqins ModelThe major simplifications involved in Equation (2D-4) are that it does not account forthe probability of overlapping chains and the volume change upon mixing of the polymer andsolvent. The volume change cannot be accounted for in a lattice model when all of the latticesites are assumed to be filled. The probability that a lattice site is filled, however, can becalculated. Huggins (1941,1942a,b,c) included in his calculations probabilities that a polymermolecule would encounter a filled lattice site. This led to a slightly different form for Equation(2D-4), but Flory (1970) states that the refinement probably is beyond the limits of reliabilityof the lattice model.Wilson (1964) modified the Flory-Huggins theory to account for the local compositionaffects caused by the differences in intermolecular forces. From these considerations thefollowing expressions for the activity coefficients are derived.

Na1) = In(X1) - In(X 1+ A 12 X 2Z) * X 2

A

J2[X 1 + A 1 2 X 2

\n(a2) - In(X2) - In(X 2+ A 21 X 1 ) - X 1

A"- A ^[X 1 + A 1 2 X 2A21X1+X2^

AA

A 1* +

21*1 *2_

(2D 14)

'

(2I>15)

Although the Wilson activity coefficient model has proven to be useful for solutionsof small molecules, it has seen very limited use for polymer solutions most likely because ofits increased complexity relative to the Flory-Huggins equation.The application of the Flory-Huggins model to liquid-liquid equilibria is discussed inSection 2F.4. Sanchez-Lacombe Equation of StateSanchez and Lacombe (1976) developed an equation of state for pure fluids that waslater extended to mixtures (Lacombe and Sanchez, 1976). The Sanchez-Lacombe equationof state is based on hole theory and uses a random mixing expression for the attractive energyterm. Random mixing means that the composition everywhere in the solution is equal to theoverall composition, i.e., there are no local composition effects. Hole theory differs from thelattice model used in the Flory-Huggins theory because here the density of the mixture isallowed to vary by increasing the fraction of holes in the lattice. In the Flory-Hugginstreatment every site is occupied by a solvent molecule or polymer segment. The SanchezLacombe equation of state takes the form

Pf

= jn

y_v - 1

I 1 " 7 | . _Lvv2f

< 2D - 16 )

The reduced density, temperature, and pressure along with the characteristic temperature,pressure, and volume are calculated from the following relationships.f=T/T*

( 2D - 17 >

T* - e Vk

(2C 18)

-

p = p/p*

(2D-19)

P- = e * / v *

(2D-20)

v = V/V* =1P

(2D-21)

V * = Nrv *

(2D-22)

where v* is the close packed volume of a segment that comprises the molecule, and e* is theinteraction energy of the lattice per site.

Costas et al. (1981) and Costas and Sanctuary (1981) reformulated the SanchezLacombe equation of state so that the parameter r is not a regression parameter, but isactually the number of segments in the polymer molecule. In the original Sanchez-Lacombetreatment, r was regressed for several n-alkanes, and it was found that the r did notcorrespond to the carbon number of the alkanes. In addition, the Sanchez-Lacombe equationof state assumes an infinite coordination number. Costas et al. (1981) replaced the segmentlength r as an adjustable parameter with z. This modification involves the same number ofadjustable parameters, but allows r to be physically significant. Thus, the model is morephysically realistic, but there have been no definitive tests to determine whether this improvesthe correlative results from the model.5. Panaviotou-Vera Equation of StatePanayiotou and Vera (1982) developed an equation of state based on lattice-holetheory which was similar to the Sanchez and Lacombe equation of state discussed above.The first major difference between the two theories is that in the Panayiotou-Vera theory thevolume of a lattice site is arbitrarily fixed to be equal to 9.75x10" 3 m3 kmol"1, while in theSanchez-Lacombe theory the volume of a lattice site is a variable quantity regressed fromexperimental data. Fixing the volume of a lattice site eliminates the need for a mixing rule forlattice sites for mixtures. In addition, a fixed lattice volume eliminates the problem of havingdifferent lattice volumes for the mixture and for the pure components. Reasonable values ofthe volume of the lattice site do not significantly alter the accuracy of the resulting equationof state. The volume should be such that the smallest group of interest has roughly the samevolume as the lattice site. Panayiotou and Vera (1982) chose the value 9.75x10"3 m3/kmol,which accurately reproduced pressure-volume-temperature data for polyethylene.The second major difference between the Panayiotou-Vera and the Sanchez-Lacombetheories is that Sanchez and Lacombe assumed that a random mixing combinatorial wassufficient to describe the fluid. Panayiotou and Vera developed equations for both purecomponents and mixtures that correct for nonrandom mixing arising from the interactionenergies between molecules. The Panayiotou-Vera equation of state in reduced variables isP!T1

=

in

^+ 1 In *1 * ^i* ~V1 - 12V1

1

- ^lf1

(2D-23)

6. Kumar Equation of StateThe Kumar equation of state (Kumar, 1986; Kumar et al., 1987) is a modification ofthe Panayiotou-Vera model that was developed to simplify the calculations for multicomponent mixtures. Since the Panayiotou-Vera equation is based on the lattice model with thequasichemical approach for the nonrandomness of the molecules in the mixture, thequasichemical expressions must be solved.For a binary system the quasichemicalexpressions reduce to one quadratic expression with one unknown, but the number of coupled

quadratic equations and unknowns increases dramatically as the number of components in themixture increases. The Kumar modification to the Panayiotou and Vera equation of stateinvolves computing a Taylor series expansion of the quasichemical expressions around thepoint where the interaction energies are zero; that is, the case of complete randomness. Thisoperation produces an explicit result for the nonrandomness factors which can then beincorporated into the derivation of the equation of state and chemical potential expression.The resulting thermodynamic expressions are cumbersome, but rely only on easilyprogrammed summations.The advantages of the Kumar equation of state are purely computational. The resultingexpressions are approximations to the Panayiotou-Vera equation of state that will reduce tothe proper forms for random conditions. Kumar et al. (1987) state that the expressions inPanayiotou and Vera (1982) differ because of errors in the Panayiotou and Vera work. TheVera and Panayiotou expressions have been shown to be correct with the methods describedby High (Chapter 5, 1990). Thus, the discrepancies between the Kumar equation of state andthe Panayiotou and Vera equation of state must occur in the approximations due to the Taylorseries expansion.7. Hiqh-Danner Equation of StateHigh and Danner (1989, 1990) modified the Panayiotou-Vera equation of state bydeveloping a group contribution approach for the determination of the molecular parameters.The basic equation of state from the Panayiotou-Vera model remains the same:PITj

=h

Vi + _z |n Vj + (qi/ri) - 1 _ 0?Vj - 12VIf|

( 2D-24)

As in the Panayiotou-Vera equation of state, the molecules are not assumed torandomly mix; the same nonrandom mixing expressions are used. In addition, as in thePanayiotou-Vera model, the volume of a lattice site is fixed and assumed to be 9.75 X 10~3m3/kmol.The major difference between the High-Danner and the Panayiotou-Vera models is thatthe molecular parameters, S11 and v*, are calculated from group contributions in the HighDanner approach. The Panayiotou-Vera formulation provide a correlation method: themolecular parameters must be determined from experimental data. The High-Danner model,however, is capable of predicting polymer-solvent equilibria given only the structure of thepolymer and solvent molecules. The molecular interaction energy parameter, e^, is calculatedfrom group interaction energies, e k k T and emm T, using the expression:1/2e6- T =Z^V VT)^nJZ^ u0!V^(Gnk a m vt? kk f Tremm,T'k m

(2D-25)

The surface area fractions of group k in component i, 0£'', is calculated from thenumber of groups of type k in component i, V^1 and the surface area of group k, Qk:

(j)^k1Qk'!? = ^AE "mQmm

(2D-26)

The molecular hard-core volume or reference volume, v*, is calculated from the groupreferences volumes, Rk, using the expression:V1,*! = a T - Ek

(2D 27)

VfRn

'

The molecular interaction energy and reference volume are a function of temperature.Group contribution values are available for these parameters at 300 and 400 K and a simplelinear interpolation is performed to find the molecular parameters at the temperature ofinterest.Group contributions for the interaction energy, ekk T, the surface area, Qk, and thereference volume, Rk, for the High-Danner model have been calculated for the alkanes,alkenes, cycloalkanes, aromatics, esters, alcohols, ethers, water, ketones, aromatic ketones,amines, siloxanes, and monochloroalkanes. If solvents and polymers of interest contain thesebuilding blocks, the thermodynamic properties can be calculated. More detailed informationconcerning the High-Danner equation of state is given in Procedure 3E.8. Oishi-Prausnitz Activity Coefficient ModelOishi and Prausnitz (1978) modified the highly successful UNIFAC (UNIversalFunctional group Activity) model (Fredenslund et al., 1975) to include a contribution for freevolume differences between the polymer and solvent molecules. The UNlFAC model uses acombinatorial expression developed by Stavermann (1950) and a residual term determinedfrom Guggenheim's quasichemical theory. Oishi and Prausnitz recognized that the UNIFACcombinatorial contribution does not account for the free volume differences between thepolymer and solvent molecules. While this difference is usually not significant for smallmolecules, it is important for polymer-solvent systems. They, therefore, added the freevolume contribution derived from the Flory equation of state, which is discussed later, to theoriginal UNIFAC model to arrive at the following expression for the weight fraction activitycoefficient of a solvent in a polymer.In Q1 = In ^l = In Q^ + In Q* + In Q™W-j

(2D-28)

The free volume contribution is given byIn0FV-- 3Cqr 1 InIn QIn1

- 1 /3 V1"1__—[*m -1J

~]VP 1 _-11 1-CVm

~ 1 /3V___—J

J [*1 - 1 ,

Here C1 is an external degree of freedom parameter for the solvent.

(2D-29)

The combinatorial and residual contributions Q c and QR are identical to the originalUNIFAC contributions.The Oishi-Prausnitz modification, UNIFAC-FV, is currently the most accurate methodavailable to predict solvent activities in polymers. Required for the Oishi-Prausnitz method arethe densities of the pure solvent and pure polymer at the temperature of the mixture and thestructure of the solvent and polymer. Molecules that can be constructed from the groupsavailable in the UNIFAC method can be treated. At the present, groups are available toconstruct alkanes, alkenes, alkynes, aromatics, water, alcohols, ketones, aldehydes, esters,ethers, amines, carboxylic acids, chlorinated compounds, brominated compounds, and a fewother groups for specific molecules. However, the Oishi-Prausnitz method has been testedonly for the simplest of these structures, and these groups should be used with care. Theprocedure is described in more detail in Procedure 3C of this Handbook.The Oishi-Prausnitz model cannot be defined strictly as a lattice model. Thecombinatorial and residual terms in the original UNIFAC and UNIQUAC models can be derivedfrom lattice statistics arguments similar to those used in deriving the other models discussedin this section. On the other hand, the free volume contribution to the Oishi-Prausnitz modelis derived from the Flory equation of state discussed in the next section. Thus, the OishiPrausnitz model is a hybrid of the lattice-fluid and free volume approaches.

E. VAN DER WAALS MODELSThe equations of state that are described in the following sections are all derived fromwhat is called the generalized van der Waals (GvdW) partition function. The GvdW model isbased in statistical thermodynamics. It is difficult to discuss this model without recourse tothe complexities and terminology used in statistical thermodynamics. The following, however,is an attempt to give a simplistic description of the fundamentals of this approach. For athorough discussion of the GvdW theory, the presentations of Sandier (1985) and Abbott andPrausnitz (1987) are recommended.The GvdW model relies on the concept of the partition function. The partition functionrelates the most probable distribution of energy states in a system of molecules to themacroscopic thermodynamic properties of that system. The energy modes can be divided intotranslational, rotational, vibrational, electronic, and attractive. The translational energy statedepends directly upon the volume (or density) of the fluid - more specifically on the freevolume. For small molecules the rotational, vibrational, and electronic modes depend only ontemperature. For large molecules, however, the rotational and vibrational modes also dependupon the density. The attractive energy of the system depends upon the intermolecular forcesbetween the molecules which in turn depends upon the density and temperature. The densityis related to the average distance of separation of the molecules; i.e., their location. Whereasthe lattice model describes the location of the molecules or polymer segments in terms ofsites on the lattice, the GvdW theory uses the radial distribution function. The radial distribution function is a mathematical expression which gives the probability of finding the centerof another molecule as a function of the distance from the center of the first molecule. It isdependent upon the density and temperature of the system. The exact form of the radial

distribution function is unknown; approximations based on assumed potential functions areused. Thus, we arrive at a partition function, Q, which is a complex function of temperature,pressure, and density. The key connection between this complex partition function and theequation of state is a relatively simple relation:P , RT [Una]I 3V J 1

( 2E-i)

It was with the above approach that the following equations of state were developed.1. Florv Equation of StateFlory et al. (1964) developed an equation of state based on a van der Waals modelgiven in reduced variables by:PvT

=

yi/3

_ J_

v 1 / 3 -1

(2E.2)

vf

where the reduced volume is given by the ratio of the volume to the reference volumeV = -^v*

(2E-3)

The reduced temperature is given by the ratio of the temperature to the referencetemperature:f = JL =T*

2v

*cRTSA?

(2E-4)

where the parameter c is a measure of the amount of flexibility and rotation that is presentin a molecule per segment, i.e., the vibrational and rotational energy states. The value of cwill be much larger for a polymer molecule than a low molecular weight molecule. Theproduct s/7 is the interaction energy of the molecule per segment. The reduced pressure iscalculated by:P = JLP*

=

2Pv

*2s/7

(2E-5)

The Flory equation of state does not reduce to the ideal gas equation of state at zeropressure and infinite volume.Flory and his coworkers derived the equation of statespecifically for liquid polymer solutions and were not concerned with the performance of theequation in the vapor phase. Poor vapor phase performance of an equation of state causesconsiderable difficulty, however, when one tries to apply the equation to higher pressure,higher temperature situations. The Chen et al. equation of state was developed in order toremedy this deficiency of the Flory equation of state.